Package smile.manifold

Class LaplacianEigenmap

java.lang.Object

smile.manifold.LaplacianEigenmap

All Implemented Interfaces:

Serializable

public class LaplacianEigenmap
extends Object
implements Serializable

Laplacian Eigenmap. Using the notion of the Laplacian of the nearest neighbor adjacency graph, Laplacian Eigenmap computes a low dimensional representation of the dataset that optimally preserves local neighborhood information in a certain sense. The representation map generated by the algorithm may be viewed as a discrete approximation to a continuous map that naturally arises from the geometry of the manifold.

The locality preserving character of the Laplacian Eigenmap algorithm makes it relatively insensitive to outliers and noise. It is also not prone to "short circuiting" as only the local distances are used.

See Also:

• IsoMap
• LLE

• References

  1. Mikhail Belkin and Partha Niyogi. Laplacian Eigenmaps and Spectral Techniques for Embedding and Clustering. NIPS, 2001.
• Serialized Form

Field Summary

Fields

Modifier and Type	Field	Description
final double[][]	coordinates	The coordinate matrix in embedding space.
final AdjacencyList	graph	Nearest neighbor graph.
final int[]	index	The original sample index.
final double	width	The width of heat kernel.

Constructor Summary

Constructors

Constructor	Description
LaplacianEigenmap(double width, int[] index, double[][] coordinates, AdjacencyList graph)	Constructor with Gaussian kernel.
LaplacianEigenmap(int[] index, double[][] coordinates, AdjacencyList graph)	Constructor with discrete weights.

Method Summary

Modifier and Type	Method	Description
static LaplacianEigenmap	of(double[][] data, int k)	Laplacian Eigenmaps with discrete weights.
static LaplacianEigenmap	of(double[][] data, int k, int d, double t)	Laplacian Eigenmap with Gaussian kernel.
static <T> LaplacianEigenmap	of(T[] data, Distance<T> distance, int k)	Laplacian Eigenmaps with discrete weights.
static <T> LaplacianEigenmap	of(T[] data, Distance<T> distance, int k, int d, double t)	Laplacian Eigenmap with Gaussian kernel.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Field Details

width

public final double width

The width of heat kernel.

index

public final int[] index

The original sample index.

coordinates

public final double[][] coordinates

The coordinate matrix in embedding space.

graph

public final AdjacencyList graph

Nearest neighbor graph.

Constructor Details

LaplacianEigenmap

public LaplacianEigenmap(int[] index,
 double[][] coordinates,
 AdjacencyList graph)

Constructor with discrete weights.

Parameters:

index - the original sample index.

coordinates - the coordinates.

graph - the nearest neighbor graph.

LaplacianEigenmap

public LaplacianEigenmap(double width,
 int[] index,
 double[][] coordinates,
 AdjacencyList graph)

Constructor with Gaussian kernel.

Parameters:

width - the width of heat kernel.

index - the original sample index.

coordinates - the coordinates.

graph - the nearest neighbor graph.

Method Details

of

public static LaplacianEigenmap of(double[][] data,
 int k)

Laplacian Eigenmaps with discrete weights.

Parameters:

data - the input data.

k - k-nearest neighbor.

Returns:

the model.

of

public static LaplacianEigenmap of(double[][] data,
 int k,
 int d,
 double t)

Laplacian Eigenmap with Gaussian kernel.

Parameters:

data - the input data.

k - k-nearest neighbor.

d - the dimension of the manifold.

t - the smooth/width parameter of heat kernel exp(-||x-y||² / t). Non-positive value means discrete weights.

Returns:

the model.

of

public static <T> LaplacianEigenmap of(T[] data,
 Distance<T> distance,
 int k)

Laplacian Eigenmaps with discrete weights.

Type Parameters:

T - the data type of points.

Parameters:

data - the input data.

distance - the distance function.

k - k-nearest neighbor.

Returns:

the model.

of

public static <T> LaplacianEigenmap of(T[] data,
 Distance<T> distance,
 int k,
 int d,
 double t)

Laplacian Eigenmap with Gaussian kernel.

Type Parameters:

T - the data type of points.

Parameters:

data - the input data.

distance - the distance function.

k - k-nearest neighbor.

d - the dimension of the manifold.

t - the smooth/width parameter of heat kernel exp(-||x-y||² / t). Non-positive value means discrete weights.

Returns:

the model.